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\title{10-725 Midterm Progress Report -- Fast Point Cloud Registration using Gaussian Processes (TA: Shiva)}

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Lagrangian with the 14 extra upper/lower bound constraints for each variable (the $\epsilon$'s are constant):
P(T) is the unconstrained original probability score.
$$
L(T,\lambda) = P(T)
+ \lambda_1 (t_x-\epsilon_0) + \lambda_2 (-t_x - \epsilon_0)
+ \lambda_3 (t_y-\epsilon_0) + \lambda_4 (-t_y - \epsilon_0)
+ \lambda_5 (t_z-\epsilon_0) + \lambda_6 (-t_z - \epsilon_0)
+ \lambda_7 (q_1-\epsilon_1) + \lambda_8 (-q_1 - \epsilon_1)
$$
$$
+ \lambda_9 (q_2-\epsilon_2) + \lambda_{10} (-q_2 - \epsilon_2)
+ \lambda_{11} (q_3-\epsilon_2) + \lambda_{12} (-q_3 - \epsilon_2)
+ \lambda_{13} (q_4-\epsilon_2) + \lambda_{14} (-q_4 - \epsilon_2)
)
$$
Gradient with respect to the 21-D parameter space (where each of the $\frac{dP}{dx}$ terms signifies the partial derivative in the unconstrained problem:
$$
\nabla P = [
\frac{dP}{dt_x} + \lambda_1 - \lambda_2,
\frac{dP}{dt_y} + \lambda_1 - \lambda_2,
\frac{dP}{dt_z} + \lambda_1 - \lambda_2,
\frac{dP}{q_1} + \lambda_1 - \lambda_2,
\frac{dP}{q_2} + \lambda_1 - \lambda_2,
\frac{dP}{q_3} + \lambda_1 - \lambda_2,
\frac{dP}{q_4} + \lambda_1 - \lambda_2,
$$
$$
t_x-\epsilon_0,
-t_x-\epsilon_0,
t_y-\epsilon_0,
-t_y-\epsilon_0,
t_z-\epsilon_0,
-t_z-\epsilon_0,
q_1-\epsilon_1,
-q_1-\epsilon_1,
q_2-\epsilon_2,
-q_2-\epsilon_2,
q_3-\epsilon_2,
-q_3-\epsilon_2,
q_4-\epsilon_2,
-q_4-\epsilon_2
]
$$

And the new parameter space is:

$T = [t_x,t_y,t_z,
q_1,q_2,q_3,q_4,
\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8},
,\lambda_{9},\lambda_{10},\lambda_{11},\lambda_{12},\lambda_{13},\lambda_{14}]
$


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